Adsorption and collapse transitions of a linear polymer chain interacting with a surface adsorbed polymer chain (original) (raw)
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We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer bbb ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents gamma1,gamma11\gamma_1, \gamma_{11}gamma1,gamma11, and gammas\gamma_sgammas which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method, for 2leble42\le b\le 42leble4, we have obtained specific values for these exponents, for various type of polymer conformations. We discuss their mutual relations and their relations with other critical exponents pertinent to SAWs on the SG fractals.
Journal of Statistical Mechanics-theory and Experiment, 2008
We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three--dimensional Sierpinski Gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modelled by two interacting self-avoiding walks (SAWs), one of them representing a 3D floating polymer, while the other corresponds to a chain adhered to one of the four 2D SG boundaries. We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the inter-chain interactions we propose two models: in the first model (ASAWs) the SAW chains are mutually avoiding, whereas in the second model (CSAWs) chains can cross each other. By applying an exact Renormalization Group (RG) method, we establish the relevant phase diagrams for b=2,3b=2,3b=2,3 and b=4b=4b=4 members of the 3D SG fractal family for the model with avoiding SAWs, and for b=2b=2b=2 and b=3b=3b=3 fractals for the model with crossing SAWs. Also, at the appropriate transition fixed points we calculate the contact critical exponents, associated with the number of contacts between monomers of different chains. Throughout the paper we compare results obtained for the two models and discuss the impact of the topology of the underlying lattices on emerging phase diagrams.
Journal of Physics A-mathematical and General, 2003
We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) Sierpinski gasket (SG) family of fractals. Each member of the 3d SG fractal family has a fractal impenetrable 2d adsorbing surface (which is, in fact, 2d SG fractal) and can be labelled by an integer bbb ($2\le b\le\infty$). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents nu\nunu (associated with the mean squared end-to-end distance of polymers) and phi\phiphi (associated with the number of adsorbed monomers), for a sequence of fractals with 2leble42\le b\le42leble4 (exactly) and 2leble402\le b\le402leble40 (Monte Carlo). We find that both nu\nunu and phi\phiphi monotonically decrease with increasing bbb (that is, with increasing of the container fractal dimension dfd_fdf), and the interesting fact that both functions, nu(b)\nu(b)nu(b) and phi(b)\phi(b)phi(b), cross the estimated Euclidean values. Besides, we establish the phase diagrams, for fractals with b=3b=3b=3 and b=4b=4b=4, which reveal existence of six different phases that merge together at a multi-critical point, whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface.
Journal of Statistical Physics, 1996
We study the problem of polymer adsorption in a good solvent when the container of the polymer-solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer b (2<~b~< or), and it is assumed that one side of each SG fi'actal is an impenetrable adsorbing boundary. We calculate the critical exponents ),j, ?l J, and ),~, which, within the self-avoiding walk model (SAW) of tile polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2 ~< b ~< 8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2 ~<b ~<80, we obtain specific values for these exponents. The obtained results show that all three critical exponents y~, ~'ll, and y.~, in both the bulk phase and crossover region are monotonically increasing functions with b. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limit b ~ or, when the fractal dimension of the SG fractals approaches the Euclidean value 2.
Surface adsorption and collapse transition of a linear polymer chain in three dimensions
Journal of Physics A: Mathematical and General, 1999
The critical behaviour of surface adsorption and collapse transition of a flexible selfattracting self-avoiding polymer chain is examined. Depending upon the underlying lattice and space dimensionality, phase diagrams that exhibit many different universality domains of critical behavior are found. We discuss these phase diagrams and the values of the critical exponents found from different theoretical methods.
Adsorption transition of a self-avoiding polymer chain onto a rigid rod
Journal of Physics: Condensed Matter, 2005
The subject of this work is the adsorption transition of a long flexible selfavoiding polymer chain onto a rigid thin rod. The rod is represented by a cylinder of radius R with a short-ranged attractive surface potential for the chain monomers. General scaling results are obtained by using renormalization group arguments in conjunction with available results for quantum field theories with curved boundaries [McAvity and Osborn 1993 Nucl. Phys. B 394 728]. Relevant critical exponents are identified and estimated using geometric arguments.
On the critical behaviour of a surface interacting linear polymer chain
Physica A: Statistical Mechanics and its Applications, 1996
A method based on a real space renormalization group transformation is developed to describe the critical behaviour of a surface interacting linear flexible polymer chain, represented by a self-avoiding walk. It is shown that a lattice model based on a central rule in which the starting point of the walk and the surface are taken to be in the middle of one cell, provides a suitable framework to study both the penetrable and impenetrable surfaces. In contrast to this, a method based on a corner rule in which the starting point of the walk and the surface are fixed to be one corner of a cell cannot describe the behaviour of a chain interacting with an impenetrable surface. The value of crossover exponent found by us for a square lattice are in agreement with those expected to be exact for both the cases.
Polymer adsorption on a fractal substrate: Numerical study
The Journal of Chemical Physics, 2012
We study the adsorption of flexible polymer macromolecules on a percolation cluster, formed by a regular two-dimensional disordered lattice at critical concentration pc of attractive sites. The percolation cluster is characterized by a fractal dimension d pc s = 91/49. The conformational properties of polymer chains grafted to such a fractal substrate are studied by means of the pruned-enriched Rosenbluth method (PERM). We find estimates for the surface crossover exponent governing the scaling of the adsorption energy in the vicinity of the transition point, φ pc s = 0.425 ± 0.009, and for the adsorption transition temperature, T pc A = 2.64 ± 0.02. As expected, the adsorption is diminished when the fractal dimension of the substrate is smaller than that of a plain Euclidean surface. The universal size and shape characteristics of a typical spatial conformation which attains a polymer chain in the adsorbed state are analyzed as well.
Adsorption and collapse transitions in a linear polymer chain near an attractive wall
Physical Review E, 2002
We deduce the qualitative phase diagram of a long flexible neutral polymer chain immersed in a poor solvent near an attracting surface using phenomenological arguments. The actual positions of the phase boundaries are estimated numerically from series expansion up to 19 sites of a self-attracting self-avoiding walk in three dimensions. In two dimensions, we calculate phase boundaries analytically in some cases for a partially directed model. Both the numerical and analytical results corroborate the proposed qualitative phase diagram.